Crawley passed 50 for the sixth time in the championship this season but three figures again eluded him as Derbyshire recovered from a poor start.
Jack Leaning scored an unbeaten 82 after Daniel Bell-Drummond made 69 and Crawley 53 but Derbyshire took 7 for 48 after tea to bowl Kent out for 285.
Ben Aitchison took 4 for 83 but Derbyshire lost Harry Came to close on 1 for 1.
Derbyshire’s decision to put Kent in was undermined by poor bowling in the first session which allowed the visitors to pass 100 by lunch.
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Two catches also went down with Crawley the batsman on both occasions and the only success for Derbyshire in the morning was the wicket of Jordan Cox who played on as he tried to leave a ball from Aitchison.
But Aitchison and the other seamers offered up to many loose deliveries with Bell-Drummond driving and clipping him for three consecutive fours in the 19th over.
Crawley was missed on 41 by wicketkeeper Brooke Guest who failed to cling on to an edge off Anuj Dal as he dived to his right and the England batsman was dropped six runs later when he drove Matt Critchley to short cover.
Bell-Drummond reached his 50 from 60 balls in the last over before lunch and Crawley completed his in the first over of the afternoon session when he swept Critchley for his seventh four.
But he added only two more runs before he aimed a cut at Dustin Melton and was caught behind.
Bell-Drummond also failed to go on when he prodded Dal to slip but Leaning emerged from a scratchy start to play with increasing assurance.
He took three successive fours off Melton and with Sam Billings, added 95 in 27 overs before Derbyshire hit back after tea,
Dal clung onto a fierce return catch to remove Billings for 36 before Guest pulled off a smart stumping when Ollie Robinson pushed forward to Critchley.
Sam Conners struck immediately with the second new ball when Grant Stewart gloved a lifter and Harry Podmore edged Aitchison to second slip.
The last three wickets fell in six balls before Stewart bowled Came in a dramatic end to the day.